You know the definition of a limit. You know the properties of limits. You can connect limits and continuity. Now use this knowledge to calculate the limits of complex functions in this lesson.
Properties of Limits
The addition property for limits
So let's talk for a minute about calculating limits. Let's look at the function f(x) = (x - 3)sin(x) + 10. Let's find the limit of this function as x approaches 3.7. To do this, let's first recall properties of limits. We have an addition property that might be useful here. This says that the limit as x approaches C of some sum of two functions is equal to the limit of each one of those functions taken separately and added together. This is our 'divide and conquer' property. We have another 'divide and conquer' property when looking at products. This says that if you want to take the limit of two functions that are multiplied together, you can take the limit of each one of those functions separately and multiply the answer together. Again, this is 'divide and conquer.'
Breaking Down the Function
If we go back to our function, f(x) = (x - 3)sin(x) + 10, and we want to find the limit as x goes to 3.7 of this function, we're going to look at these pieces individually. We're going to use the product rule to separate x - 3 from sin(x), we're going to use the addition rule to look at that + 10, and we're going to use the subtraction rule, which is just like the addition rule, to separate the x - 3 into x and 3. Now I'm pretty confident that if I graph out x and 3, I can show that as x goes to 3.7, x will also go to 3.7. As x goes to 3.7, 3 will go to 3. I'm also pretty confident that as x goes to 3.7, 10 will stay at 10. So I can plug in all these numbers except for this sin(x). So what do we do about that?
Using the product, addition, and subtraction rules to break down the function
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It turns out that there's one other useful rule for finding limits. For all polynomials and rational functions - and even trig functions and square roots - if the function is defined at the limit, the value of the function at that limit is equal to the limit itself. So what does this mean? This means that the limit as x goes to some number (like 3.7) of some function is equal to the value of that function at that number as long as the function is a polynomial, rational function, trig function or square root, where this is defined. If I go back and ask what is the limit as x goes to 3.7 of sin(x), I know that sin(x) is defined everywhere, so I can just find sin(3.7), which is about -0.5 (remember to use radians). So if I plug all of these things into my original function, then I know that the limit as x goes to 3.7 of my function f(x) can be split up, using all the rules that we know, and calculated out to be roughly 9.63.
Why is this the case? Why is it that sometimes you can write the limit as x goes to some number like C of f(x) is equal to f(C)? And more importantly, when can you not use this rule? Well, the trick is in continuity. All of these functions - the trig functions that are defined, the square roots when they're defined, polynomials, rational functions - are continuous functions. When you have a continuous function, the limit of that function, as you approach some number like C, equals the value of that function at C. So this is just what we said before: the limit as x goes to C of f(x) equals f(C). There's no jumping here. If you graph it, your finger always stays on the paper. So it makes sense that as you approach some number, you're going to hit that number; there's no discontinuity.
The process of solving the continuous function example
If I look at a function like f(x) = x^3(cos(x+3))^2, I know that this is a continuous function. I can graph it. I also know that everywhere this function is defined. So I know that if I want the limit of f(x) as x goes to 9.1 of this function, I can just plug in 9.1 for x (remember to switch your calculator to radians). When I do that, I calculate that the limit of f(x) as x goes to 9.1 is roughly 601.2.
To recap, the easy way to find limits: For continuous functions, use substitution for finding limits. What I mean by this is if f(x) is continuous, then the limit as x goes to C of f(x), is equal to f evaluated at C.
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